| L(s) = 1 | + 2.54·3-s + 1.76·5-s + 3.49·9-s + 3.24·11-s + 0.155·13-s + 4.49·15-s + 0.784·17-s − 6.00·19-s + 3.80·23-s − 1.89·25-s + 1.25·27-s − 1.80·29-s + 4.59·31-s + 8.27·33-s + 0.137·37-s + 0.396·39-s + 3.05·41-s + 12.5·43-s + 6.16·45-s − 6.72·47-s + 1.99·51-s + 12.3·53-s + 5.72·55-s − 15.3·57-s + 13.8·59-s + 6.42·61-s + 0.274·65-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.788·5-s + 1.16·9-s + 0.979·11-s + 0.0431·13-s + 1.16·15-s + 0.190·17-s − 1.37·19-s + 0.792·23-s − 0.378·25-s + 0.242·27-s − 0.334·29-s + 0.824·31-s + 1.44·33-s + 0.0225·37-s + 0.0634·39-s + 0.476·41-s + 1.91·43-s + 0.918·45-s − 0.980·47-s + 0.280·51-s + 1.69·53-s + 0.772·55-s − 2.02·57-s + 1.80·59-s + 0.822·61-s + 0.0340·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.391859572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.391859572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2.54T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 13 | \( 1 - 0.155T + 13T^{2} \) |
| 17 | \( 1 - 0.784T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 - 0.137T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 6.42T + 61T^{2} \) |
| 67 | \( 1 + 6.39T + 67T^{2} \) |
| 71 | \( 1 + 3.65T + 71T^{2} \) |
| 73 | \( 1 - 9.95T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 - 0.163T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.312867626785740725267069031971, −7.58136969994813024900303291029, −6.76168418347052749961214899389, −6.16515065459210448513004003735, −5.26362079584010886851972522339, −4.14338576407482257284170430754, −3.73719977511457760084300380033, −2.59501335097539130093241722153, −2.16352679631667551760745376644, −1.11751731995120687719806507075,
1.11751731995120687719806507075, 2.16352679631667551760745376644, 2.59501335097539130093241722153, 3.73719977511457760084300380033, 4.14338576407482257284170430754, 5.26362079584010886851972522339, 6.16515065459210448513004003735, 6.76168418347052749961214899389, 7.58136969994813024900303291029, 8.312867626785740725267069031971
